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Universit`

a degli Studi di Pisa

DIPARTIMENTO DI FISICA Corso di Laurea Magistrale in Fisica

Tesi di laurea magistrale

Quantum Quench in one-dimensional gases

Candidato:

Paolo Pietro Mazza

Relatore:

Prof. Pasquale Calabrese

Correlatore:

Dott. Mario Collura

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PREFACE

I

n this work we discuss the relaxation properties of a quantum one dimen-sional gas. The interest in studying this kind of systems has recently grown due to the development of innovative experimental techniques that made pos-sible the confinement of particles in quasi one-dimensional optical lattices weakly interacting with the environment. These properties can be explored using a quan-tum quench: we prepare the system in the ground state of a given hamiltonian, then we suddenly change a parameter and let it evolve unitarly. There are mainly two types of quantum quenches: local and global. We focus our attention on a global quantum quench and we see how the relaxation occurs as a many body effect.

The structure of the thesis is the following.

In the first chapter of this thesis there is a brief overview about the recent theoret-ical developments about thermalization of isolated quantum systems. In partic-ular the attention is focused on the role of the spatial dimensionality and on the conservation laws. In fact it is widely believed that one dimensional integrable systems relax toward a non-thermal distribution that, in a certain sense, retains “more memory” of the initial conditions than a thermal one.

In the second chapter some notions about quantum integrability are given. We review the concept of classical integrability and we illustrate some criteria for the quantum case. This is useful in the classification of the models between integrable and non-integrable ones.

In the third chapter the model of interest is deeply analyzed. We deal with a many body hamiltonian with point-like interactions. This model, for general values of the coupling constant, is solvable via Bethe Ansatz techniques. How-ever we are interested in two limiting cases: free bosons (no interaction) and Tonks-Girardeau limit, which is often referred as “hard-core” (HC) bosons gas. We show that the latter model is exactly mappable to a spinless free fermionic gas using a Jordan Wigner transformation. Furthermore we show some results from a problem in which the interaction parameter is quenched between c = 0 and c = ∞ (this corresponds to a quench from Bose-Einstein condensate to HC bosons), with periodic boundary conditions.

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iii to this thesis. We consider a global quantum quench in a confined one dimen-sional bosonic gas. This configuration is very interesting from an experimental point of view. We show that relaxation occurs in slightly different manner than in the periodic case: the stationary correlation function “feels” the boundaries also in the thermodynamic limit. Furthermore we find a compact expression for the time dependent density profile and for the fermionic correlation function. Both functions describe the non equilibrium behavior of the system. The solutions of the confined problem present difficulties which were absent in the periodic case. These have been overcome by some ingenious approximations which become ex-act in the thermodynamical limit, thus providing the analytical solution to the problem. In the course of the computation numerical analysis is often used as a support.

We found out that the long-time state of the confined system is translation-ally invariant (we demonstrated that non translationtranslation-ally invariant corrections are finite-size effects), in particular the stationary density profile is the same as in the homogeneous case, as naively expected. But the effects of the confinement are visible both in the stationary two point correlation function and in the non trivial evolution of the density profile.

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CONTENTS

1 Introduction 3

1.1 Non equilibrium dynamics . . . 6

1.2 Experimental results . . . 8

1.3 The Generalized Gibbs Ensemble . . . 13

1.3.1 Locality . . . 16

2 Integrability 18 2.1 Classical Integrability . . . 18

2.1.1 Integrability and ergodicity . . . 20

2.2 Quantum Integrability . . . 21

3 The Lieb-Liniger model 25 3.1 The model . . . 25

3.2 Tonks-Girardeau limit . . . 29

3.2.1 Jordan-Wigner transformations . . . 30

3.3 Quantum quench from free to hard-core boson in a periodic 1-D gas 32 4 Quantum quench in a confined 1-D gas 36 4.1 The quench protocol . . . 37

4.2 Generalized Gibbs Ensemble results . . . 38

4.2.1 Initial fermionic correlation function . . . 40

4.2.2 Some general features about quantum-quenches in trap . . 41

4.2.3 Fermionic mode occupation . . . 44

4.2.4 Density profile . . . 46

4.2.5 Two-points correlation function . . . 48

4.3 Temporal evolution of the density profile . . . 51

4.4 Time-dependent two-points correlation function . . . 58

A 63 A.1 Setup on lattice . . . 63

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CONTENTS 2

B 67

B.1 Demonstration of equation (4.49) . . . 67 B.2 Approximation used in the dynamical correlation function . . . 68

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CHAPTER

1

INTRODUCTION

O

ne of the hardest challenges in physics is the study of systems com-posed by an enormous number of particles. The difficulty in dealing with these kinds of systems is that it is practically impossible to get (and store. . . ) informations about the positions and momenta of all the parti-cles, and even if this were possible, it would be impossible (or very hard. . . ) to solve the equations governing its dynamics.

These obstacles can be circumvented by introducing macroscopical quantities that describe the collective effect of the interactions between the microscopical constituents of the system. Examples of macroscopical quantities are: tempera-ture, pressure, etc. . . . The words “collective” and “macroscopical” mean that it is possible to describe the state of the system in this simplified view only when we deal with a huge number of particles that together constitute a macroscopic system (i.e. it makes little sense to ask the temperature of a system with one particle). One fundamental assumption behind thermodynamics and classical statistical physics is that of thermal equilibrium: let us image two systems A and B with different temperatures TA and TB. If we put them in contact after a

transient time the whole system (A + B) will reach a stationary state character-ized by an equilibrium intermediate temperature TA+B; we say that the system

thermalized. Statistical physics and thermodynamics do not tell us how the ther-malization occurs but they fully describe the equilibrium state. The thermal state reached after the thermalization has a well defined temperature and then a well defined total energy, this steady state can be described and fully analyzed using the “Gibbs ensembles”, based on the hypothesis that the time average coincides with the average over the ensemble. Let us briefly summarize some properties about Gibbs ensembles

The microcanonical ensemble 1 is a set of points in the phase space of a

classi-1Other ensambles can also be defined, i.e. the canonical and the gran-canonical. These are

particularly useful when we want to analyze the thermal state of subsystems in a thermal bath. The energy and, in the grancanonical case the number of particles too, of the subsystem are not fixed, although the quantities related to the entire system are fixed. Differences between ensembles are present only for finite systems. When the thermodynamical limit is considered,

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CHAPTER 1. INTRODUCTION 4 cal system such that each point is characterized by a fixed total energy(namely the energy belongs in a small interval [E and E + ∆E]). Each point of this set defines a possible representative state for the system we are considering. From a macroscopical point of view all these states are indistinguishable, therefore all the points in the ensemble are equiprobable. When the system thermalizes, it can be found in any of those microscopical configurations. The ergodicity hypothesis states that, if the system is not integrable, it “explores” all the possible allowed micro-states in the phase-space and therefore the average over the the ensemble coincides with the time average. Integrable or quasi integrable systems present periodic or quasi periodic trajectories in the phase space and therefore the ergodic hypothesis is broken2.

So far we have considered classical systems. In the quantum case the situation is more complicated. The most important difference in this case is the impossibility to define a phase-space and therefore the impossibility to translate the ergodic theorem from classical to quantum mechanics. Anyway this impossibility does not imply that a quantum statistical system has to be necessarily described by solving exactly its dynamics. It is possible to define the Gibbs ensemble also in this case introducing the density matrix representation. The density matrix representation is a formal procedure that it is particularly suitable when we deal with systems in which there is a lack of information that does not permit to con-struct a wave function for the entire system [1]. The averaging procedure using the density matrix has a twofold nature, it considers both the averaging over the intrinsic probabilities of the quantum mechanical description and the averaging due to the incompleteness of information of the whole system analyzed. Pure sys-tems can also be described using density matrix representation, in this case only the probabilistic average over the quantum nature of the system is considered. Before proceeding we must understand under which hypotheses the characteriza-tion of a thermal quantum state through the ensembles technique makes sense. Let us suppose to have an initial state described by the wave function:

|Ψ(0)i =X

n

cn|ϕni, (1.1)

where |ϕni are stationary eigenfunctions of a certain Hamiltonan. The time

evolved state is:

|Ψ(t)i =X

n

e−iEn

~ tcnni (1.2) If we consider a certain observable A, its expectation value at the generic time t is: hΨ(t)|A|Ψ(t)i =X n,m cm(t)cn(t)hϕm|A|ϕni, (1.3) where cn(t) = e−i En

~ tcn. If a steady state exists, it must coincide with the time average over an infinite interval of time. The stationary expectation value of the

the characterization of the steady state is equivalent in the three pictures and the distinction becomes only formal.

2

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CHAPTER 1. INTRODUCTION 5 observable A is therefore:

hAi=X

n,m

cm(t)cn(t)hm|A|ni (1.4)

with cm(t)cn(t) = cmcnei(Em−En)t= |cn|2δn,m.

Relation (1.4) holds only for an isolated system, if we consider a system in thermal contact with a bath the former relations are not valid. In order to assure the existence of a equilibrium state we must postulate that

cm(t)cn(t) = |cn|2δn,m. (1.5)

In statistical physics this postulate is known as random phases postulate, and states that a quantum subsystem in thermal equilibrium with a bath is an in-coherent superposition of the eigenstates of a certain Hamiltonian. The physical meaning of this postulate is that the time average cancels all the quantum inter-ferences among the states during the measure procedures. This is what is usually referred as a “statistical mixture”. Under the assumption of the random phases postulate the value of a certain observable can be written as:

hAi= Tr(ρA), (1.6) where ρ =P

n|cn|2|ϕnihϕn|is the density matrix of the mixed state.

Let us consider an example that will clarify a bit why the random phases postulate is physically reasonable. Let us take a system S with two energy eigenstates (|1iS

and |2iS) in thermal equilibrium with the environment E that is in the state |Ei0,

the initial state of the composed system S + E is described by the wave function: 0iS+E = (α|1i + β|2i) ⊗ |Ei0= α|1, E0iS+E+ β|2, E0iS+E. (1.7) If we suppose to have a weak interaction between the environment and the system of the type:

Hin = C1|1ih1| ⊗ V1+ C2|2ih2| ⊗ V2

after a time interval the state will evolve in

tiS+E = α|1, Et(1)iS+E+ β|2, Et(2)iS+E. (1.8) Let us note that the eigenstates of the system S are stationary instead the en-vironment evolved in different way according to the different coupling with the eigenstates of the subsystem. The reduced density matrix of the system S is

ρS = ρ11|1iSh1| + ρ12|1iSh2| + ρ∗12|2iSh1| + ρ22|2iSh2|, (1.9)

the matrix elements can be calculated using the definition of reduced density matrix TrE[|φtiS+Ehφt|]:        ρ11= |α|2 ρ22= |β|2 ρ12= αβhEt(1)|E (2) t i . (1.10)

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CHAPTER 1. INTRODUCTION 6 At this point we can make some observations. The environment has a lot of degrees of freedom, by hypothesis it is much larger than the system S, therefore assuming casual time evolution the states are coupled randomly then it is highly probable that hE(1)

t |E (2)

t i  1, this means that the reduced density matrix is

practically diagonal.

t For isolated quantum systems the situation is a bit different, in fact if the system is prepared in a pure state:

ρin = |ΨihΨ|, (1.11)

the unitary time evolution due to the Schroedinger equation:

|Ψ(t)i = eiHt~ |Ψ(0)i, (1.12) will maintain it in a pure state all time. Therefore thermalization in this case can not be reached in the sense seen formerly because the behavior could display periodicity in time. For isolated quantum systems, then, the limit

lim

t→∞|Ψ(t)i (1.13)

can be studied only passing through the exact non-equilibrium dynamics of |Ψ(t)i. Actually, as we shall see later, with a suitably choose of the order of limits, an infinite subsystem of a certain isolated system can thermalize, or, said in another way, it acts as its own thermal bath [3].

1.1

Non equilibrium dynamics

In recent years a great theoretical effort has been devoted to the study of the unitary time evolution of isolated quantum systems. The main questions that we are trying to answer are:

• What are the characteristics of the steady state of an isolated quantum system?

• When this state is described by a thermal state with a effective temperature Tef f?

• What is the role played by the spatial dimensionality and the conservation laws? Do they affect the characterization of the steady state?

These questions arose for the first time in 1929 (Von Neumann et al. [4]) but be-fore the last decades they were considered of purely academic interest. Recently the development of new experimental techniques have made possible the direct observation of non-equilibrium dynamics of isolated quantum systems. A fruit-ful comparison between theory and experiments has been possible thanks to the use of cold atomic gases and nanostructures. In particular it has been possible to construct quasi one-dimensional optical lattices that are so weakly interacting with the environment that their evolution can be considered unitary. Therefore it has been possible to see if, and eventually how, the spatial dimensionality affects the evolution of these systems. We refer to the section (1.2) for a more detailed

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CHAPTER 1. INTRODUCTION 7 discussion of the experimental results.

The interest in understanding the out-of-equilibrium behavior of quantum inter-acting systems is not only important from a fundamental point of view, indeed there could be also technological implications. A quantum computer is the most remarkable field of application of these topics, in fact it will definitely require the capability of performing real time manipulations of interacting quantum sys-tems. Therefore it is of crucial importance to understand the coherent dynamics of these systems since it could be one of the main points of various experimental setups and of future technologies.

There are several ways in which a system can be driven out of equilibrium. For example, it is possible to prepare a system in a equilibrium state and then apply a driving field, or pumping energy and particles. One of the most used technique both in theoretical and experimental field is that of the quantum quench: one prepares a quantum system in the ground state of a certain hamiltonian Hbq

with a tunable parameter; at t = t0 we change this parameter suddenly, the time

evolution will be given therefore by another post-quench hamiltonian Hpq, this

corresponds to an out-of-equilibrium evolution. It is important to remark that the changing in the parameter must be done instantaneously, in practice this means in a time interval ∆t  ∆tevolutionwhere ∆tevolution is the minimum time

scale of the system.3In the opposite limit, we would explore an adiabatic regime.

There are mainly two types of quenches: global quenches and local quenches. In a local quench the change in the hamiltonian is localized, for example impurities on lattices, in the global quenches the hamiltonian is changed over the whole system. In this work we analyze a global quench; this means that we consider a certain hamiltonian (namely Lieb-Liniger model, see chapt. 3) with an interaction term that will be quenched between two different values of the coupling constant. From a theoretical point of view the quench problems can be tackled in different ways:

• Numerically, using e.g. DMRG techniques or exact diagonalization; • Analytically, exploiting various techniques (e.g. solving free theories after

an exact mapping, or via Bethe Ansatz. . . )

In this work only analytical techniques are used. Numerical analysis has been only a support in order to understand the behavior of some quantities which were hard to treat analitically.

Before proceeding, it is of fundamental importance to point out some things about the infinite time limit for closed systems [5]. In fact for finite systems the t → ∞could not exists due to the effect of quantum revivals of matter waves. In order to bypass this obstacle it is sometimes convenient to consider time averaged quantities. This problem disappears when we work in the thermodynamic limit (TDL), that is when we consider a system composed by an infinite number of particles N lying on an infinite volume L such that the density n = N

L is constant,

the large-time limit must be taken after the TD one.

The revivals of matter waves have been observed in one of the first experiments

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CHAPTER 1. INTRODUCTION 8

Figure 1.1: Figure from [6]. Interference patterns for different “hold times”: (a) τ = 0 µs, (b)

τ = 100 µs, (c) τ = 150 µs, (d) τ = 250 µs, (e) τ = 350 µs, (f) τ = 400 µs, (g) τ = 550 µs. We note that after a hold time of 250µs the information about the initial pattern is completely lost, but it is restored after τ = 550µs, this a classical example of matter wave revival.

about the out of equilibrium dynamics of closed systems performed by Greiner et al. in 2002 [6]. In this experiment it has been considered a Bose gas in a three dimensional cubic lattice, in this configuration the system is well approximated by a Bose-Hubbard model described by the hamiltonian

H= −JX hiji (aiaj+ aiaj) + U 2 X i ni(ni1), (1.14)

where ai are bosonic operators and ni = aiai are the number operators on site.

The system is initially prepared in a “superfluid phase”, that is, it is prepared in the ground state of hamiltonian (1.14) with J  U, in which the tunneling terms dominate. The on site interaction term U is suddenly quenched such that U  J, the time evolution is performed by the hamiltonian:

HpqU 2 X i ni(ni1). (1.15)

The system is “hold” and left evolve for several time intervals and then is released from the trap. Absorption images of interference patterns after the release are shown in figure 1.1. As we can see after a time τ = 550 µs the system returns in its initial configuration. This experiment shows that it is possible to maintain a system isolated for a time sufficient to show its coherent dynamics.

1.2

Experimental results

In this paragraph some of the most important experimental results, that encour-aged theoretical interest in the arguments previously outlined, will be presented. The first one that we will discuss is the quantum Newton’s Cradle [7] performed by T. Kinoshita, T. Wenger and D.S. Weiss. In this experiment it has been ob-served the non-equilibrium dynamics of a 1D Bose gas composed by87Rb atoms.

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CHAPTER 1. INTRODUCTION 9 2D dimensional lattice, here, using a red-detuned crossed dipole trap, the bidi-mensional lattice is divided in arrays of 1D Bose gases weakly interacting. The dynamics in each tube is strictly one dimensional because the weak transverse excitation energy (Et= ~ωr, ωr = 67kHz) far exceeds the thermal energies of the

trapped atoms and therefore the tunneling probability is very low.

Figure 1.2: Figures from [7]. Left: Schematic representation of the atomic bundles in the 1D

harmonic trap. Right: Absorption images in the first oscillation cycle. The dynamics is always 1D since the collision energy of the bundle is less than a quarter the transverse confinement energy.

Using two shortly separated pulses the atoms in every array are put in a super-position of momentum states with p = ±2~k, in this way each bundle is split in two atomic clouds with opposite momenta. The system is left evolving, the bundles collide in the centre of the trap twice a cycle (Fig. 1.2), the collision energy is still less than a quarter of the minimum transverse excitation energy, ensuring that the dynamic remains 1D. There is not perfect recurrence due to dephasing effects. It is possible to extract also the momentum distribution. As we see in Fig 1.3, also after several periods of oscillations the distributions are far from being gaussian meaning that there is not thermalization.

These distributions, in a certain sense, carry more informations about the initial conditions. The same experiment has been done for 2D and 3D Bose systems (see Fig. 1.4) , in that case thermalization occurs after a few periods. Therefore the spatial dimensionality plays a crucial role in the thermalization of isolated quantum systems.

The next experiment we are going to describe is conceptually slightly different from the previous one, but is important to show how nowadays it is possible to manipulate microscopic systems in order to implement quantum quenches and then study the non-equilibrium dynamics [8]. In this experiment the expansion of a fermionic gas in a 2D lattice after a trap release is studied. Two cases are examined: interacting and non-interacting fermions (see Fig. 1.5)

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CHAPTER 1. INTRODUCTION 10

Figure 1.3: Figures from [7]. Left: Momentum distribution for t > 1910τ , with τ oscillation

period, and γ = 3.2. Right: Momentum distribution for t > 390τ , with τ oscillation period, and

γ = 18. γ = |2/a1Dn1D| where a1D is the 1D scattering length and n1D the average density.

The red curves are real data taken in a certain tobs, the blue and green curves are simulations from different models that take into account the losses during the measure. For details see [7].

Figure 1.4: Momentum distribution of a 3D Bose gas in the same conditions described in [7].

t = 0τ , t = 2τ , t = 4τ , t = 9τ , with τ oscillation period, we can see that thermalization occurs after a few periods.

This system is well described by a Fermi-Hubbard model with an external har-monic potential that breaks the horizontal translational invariance:

H= −J X hiji,σ (ci,σcj,σ+ h.c.) + U X i

ni,↑ni,↓+ Vext., (1.16)

where the ci are fermionic operators and σ are spins degrees of freedom. The

system is initially prepared in a confined state, due to a tight harmonic trap, and then is released. This is an example of “inhomogeneus” quench: an external pa-rameter of the hamiltonian is changed inhomogeneously along the whole system. Also these types of quenches have been studied from a theoretical point of view ([9]).

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in-CHAPTER 1. INTRODUCTION 11

Figure 1.5: Figure from [8]. Schematic representation of the experiment. After the release

of the harmonic trap, the in-situ density is studied both in the interacting case and in the non-interacting case, significative differences result.

Figure 1.6: Figure from [8]. Left: time evolution of the density for non-interacting fermions.

For large time (i.e. in the last panel) the system is homogeneous. Right: Density distributions 25 ms after the release for several values of the on-site interaction (both repulsive and attractive). The system is rotationally invariant especially for high values of the interaction (first and last panel).

situdensity (see Fig.1.6). For non-interacting fermions the evolution is given only by the hopping term of (1.16), (Hpq = −JPhiji,σ(c

i,σcj,σ+ h.c.)), the expansion

is ballistic, that is, every excitation expands independently with constant quasi-momentum. This results in a square-shaped density distribution at the equilib-rium. In the interacting case the situation is much more complicated, in this case the rotationally invariance of the density shape is preserved and the expansion is diffusive.

The next, and last, experiment that we are going to show was performed by Trotzky et al. in 2012 [10], they studied the relaxation properties of a one-dimensional Bose gas.

In this experiment a Bose Einstein condensate composed by87Rb atoms is loaded

in a three dimensional lattice and using a crossed trap is divided in an array of 1D chains. According to the data, in the 3D lattice about 45 · 103 atoms are loaded.

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CHAPTER 1. INTRODUCTION 12 measured quantities have to be intended as an average over all the chains. Once that the 1D systems are created a “short lattice” with wave length λsl = 765nm

is added along the direction of the chain with relative phase adjusted with the trapping lattice to load the sites of the short lattice alternately (for example only on “even” sites, see [10] for the technical details), as shown in the first picture of the Fig. 1.7.

Figure 1.7: Figure taken from [10]. Schematic representation of the three steps of the

experi-ment.

This system is well described by a Bose-Hubbard model with the same Hamil-tonian (1.14) and everything is set in such a way that the tunneling probability is very low. Tuning the parameters, the potential is suddenly changed and the tunneling between adjacent sites is activated, the system is left evolving for a certain time t. After this time interval the potential in changed again and the tunneling is blocked, the average population on “odd” sites is then measured. This process is repeated for several values of t and four values of the ratio J

U.

Figure 1.8: Figure taken from [10]. Evolution of the average population of the odd sites

after the quench, the dots are the experimental data, the blue lines are results from numerical simulations.

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CHAPTER 1. INTRODUCTION 13 sites, that were empty in the initial state, after a transient period, approaches a constant stationary value. This fact agrees with numerical simulations performed using t-DMRG techniques.

The experiments discussed in this section are only a small sample of all the works done in the study of the non-equilibrium dynamics of isolated systems, other im-portant examples, that we will not discuss here, can be found in references [14], [13], [12] and [11].

1.3

The Generalized Gibbs Ensemble

From the results of the experiments described in the former paragraph (especially [7] and [10]) it seems that isolated quantum systems prepared in a pure states, in the limit t → ∞ after a quench, present characteristics typical of mixed states. As we have discussed in paragraph 1.1 systems prepared in a pure state evolve unitarily according to the Schroedinger equation, therefore they can never reach a steady state described by a density matrix. How can this apparent paradox be solved? The solution [15] is that in the thermodynamical limit finite subsystems reach an equilibrium steady state well described by a reduced density matrix that has all the characteristic of a mixed state. If we consider a system in a pure state |Ψ(t)i, and a subsystem A, its reduced density matrix can be obtained tracing out the degrees of freedom of the complementary system A:

ρA(t) = TrA(|Ψ(t)ihΨ(t)|). (1.17)

The expectation value of an operator OAwhich degrees of freedom are within

the subsystem A reaches an equilibrium value that can be found from (1.17): lim

t→∞hΨ(t)|OA|Ψ(t)i = Tr[ρA(∞)OA]. (1.18)

Therefore ρA(∞) is effectively a density matrix describing the equilibrium

prop-erties of a mixed state. If it happens that for any finite subsystem A of our system the limit:

lim

t→∞ρA(t) = ρA(∞) (1.19)

exists we can make the following argument [15]: first of all, our entire system is decomposed in a subsystem A and its complementary A, at this point we can take the thermodynamical limit of the system keeping A finite. Then, finally, we can make the system A very large4 itself. Since relation (1.19) holds for every

subsystem it will continue to be valid also for a system A built in this way. Then, if the order of the limits is respected, the subsystem A will reach an equilibrium state and all the operators within it will attain a value well described by a density matrix. Suggestively we can say that the system acts as its own bath.

The experiment previously described (Fig. 1.4) suggests that the reduced density matrix of the subsystem is thermal for 2D or 3D systems. In the configuration used in that case (a harmonic confining potential) a thermal density matrix is

4

Strictly speaking the system A at this point can be considered practically infinite but still smaller than the entire system.

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CHAPTER 1. INTRODUCTION 14 recognized by having a gaussian momentum distribution at the equilibrium. In this case we have only few conserved quantities (i.e. the energy and the number of particle) and the steady state can be described by a Gibbs ensemble with an effective temperature Tef f.

For one-dimensional systems the situation is a slightly different, as we have seen (fig. 1.3), and as conjectured by Rigol et al. [16] [17] [18] [19], for integrable quantum systems there is relaxation to a non-thermal distribution. Quantum integrable systems are characterized by an infinite set of higher conserved charges {Qi} that seem to play a key role in the description of the stationary state [20] [21] [22].

In [16] a model very similar to that considered in this thesis, but in the discrete case, it is studied: a one-dimensional lattice of length L loaded with hard-core bosons. This model is fully described by a Bose-Hubbard hamiltonian with no on-site interaction5. ˆ H= −J L X i=1 (ˆbiˆbi+1+ h.c.), (1.20) where [bi, bj] = [bi, bj] = [bi, b

j] = 0 as long as i 6= j and {bi, bi} = 1, with

(bi)2 = (bi)2 = 0 for all i. This model can be mapped to a free fermion

the-ory (see paragraph 3.2.1) and therefore it is exactly solvable. In [16] the au-thors questioned, assuming that after an out of equilibrium dynamics this system reaches a stationary state, what is the statistical many-body density matrix that would have described it. In order to answer this question it was conjectured that standard prescriptions from statistical mechanics apply, therefore it should be maximized the many-body entropy taking into account the constraints imposed by all the integrals of motion. This resulted in the Generalized Gibbs Ensemble (GGE): ρGGE = e −P mλmIm Tr(e−P mλmIm) , (1.21)

where {Im} is a full set of local (see paragraph 1.3.1) integrals of motion, and

the λm are Lagrange multipliers fixed by initial conditions via:

Tr(ImρGGE) = hImit=0. (1.22)

From (1.22) it is evident that the GGE carries an infinite amount of information about the initial conditions, this could explain why the equilibrium momentum distribution is not single peaked, as in the gaussian thermal case, but maintains a shape that in a certain sense “remembers” the initial distribution (this concept will be clearer soon). The integrals of motion {Im}are defined as operators such

that:

[ ˆH, Im] = 0,

where ˆH is the hamiltonian of our model. Therefore, being conserved quantities, they are the most intuitive choice to do if we want to catch the stationary state properties.

5

This was the same system studied in experiment [7], it was this experiment that motivated Rigol et al. to investigate theoretically the thermalization properties of this system.

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CHAPTER 1. INTRODUCTION 15 In order to understand why the generalized Gibbs ensemble has that form we can use the “maximum entropy argument” made by E.T. Jeynes [23]. In that article, information theory is exploited to define a rigorous criterion that could help in the construction of statistical ensembles when we have some constraints (e.g. many conserved quantities). Quoting from article [23]: “Previously one constructed a theory based on equation of motion, supplemented by additional hypotheses of ergodicity [. . . ] and the identification of the entropy was made at the end [. . . ] Now, however, we can take entropy as our starting concept, and the fact that a probability distribution maximizes the entropy subject to certain constraints becomes the essential fact which justifies the use of that distribution for inference. [. . . ] we make it possible to see statistical mechanics in a much more general light”.

In order to clarify slightly more this last concept let us consider a physical system with n  1 states, to each state corresponds a certain probability pi such that

P

ipi = 1. Let us suppose that we know r  n expectation values of the operators

O(l):

E(l)=X

i

pihφi|O(l)|φii,

what is the most probable expectation value of another operator F(x) that cannot be decomposed in terms of O(l)? We would like to find it using constraints

that implement the little informations we have. The only physical quantity that measures uncertainty, increasing with increasing uncertainty, that is positive and additive can be demonstrated to be the Shannon’s entropy, defined as:

S({pi}) = −

X

i

piln(pi). (1.23)

Note that this quantity coincides with the Gibbs’ entropy. If we know nothing about the expectation values, the entropy is maximized if pi = n1 ∀i, this is

exactly the microcanonical ensemble. If we, instead, have a constrain only on one expectation value (e.g. the energy) E = P

ipihφi|O|φii, entropy can be

maximized introducing a the Lagrange’s multiplier β and this gives us the well-know canonical ensemble

pi = Z−1e−βAi. (1.24)

The same argument can be exploited when we have many constraints (as in the GGE . . . ). In this case we must use several Lagrange’s multipliers βl , one for

every known expectation value, this, if the Ai commute leads to

pi = e

−Pr

l=1βlAl

Z , (1.25)

where Z is a normalization factor (i.e. the partition function). This form is very similar to the (1.21). Let us return to the discussion of Ref. [16]: the predictive power of the generalized Gibbs ensemble is tested using numerical techniques. The system is prepared in the ground state of a spatially-periodic background-potential with period 4; Vext. = APicos

 2πi T bibi 

, then is released (Vext. = 0)

to a flat-bottom hard-wall box. In the paper, numerical simulations are used to study the exact out-of-equilibrium dynamics and the steady state attained.

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CHAPTER 1. INTRODUCTION 16 The data are then compared with the results predicted by the Generalized Gibbs Ensemble.

Figure 1.9: Figure taken from [16]. Upper : time evolution of the quasi-momentum distribution.

Lower : distribution of the quasi-momentum after relaxation. The discrepancy between the results of the time evolution and the prediction of the GGE ensemble are less than the width of the line (see [16] for further details).

As it can be seen in figure 1.9 there is a perfect matching between the GGE predic-tions and the results from dynamical evolution, indeed the lines are completely overlapped. From Fig. 1.9 we can understand why the generalized stationary state carries “more memory” of the initial conditions, in fact also in the equilib-rium state can be observed well-separated peaks in the momentum distribution similar to the initial one.

1.3.1 Locality

In the last paragraph it has been stated that the generalized Gibbs ensemble is built using a complete set of integral of motion Im. An important point to

face concerns what kind of operators should be included in the definition (1.21). Any quantum system has too many integrals of motion; the easiest examples are the projectors onto the energy eigenstates In = |ψnihψn| with H|ψni= En|ψni,

since [H, In] = 0. However the projectors can not have any role in describing

the properties of a system after the thermalization. A possible solution to this dilemma has been conjectured by P. Calabrese, F.H.L. Essler and M. Fagotti in [15] and [24]. Since we focus on local properties of systems they proposed to include only local integrals of motion. These integrals can be written as sums (or, in the continuum case, as integral) of operators acting locally, In = R dxJ(x),

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CHAPTER 1. INTRODUCTION 17 with a procedure that remembers the Noether’s theorem. This conjecture has demonstrated to work in many cases , but the proposal is still under debate (see [25][26][27][28][29][30][31][32][33] for some criticism).

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CHAPTER

2

INTEGRABILITY

I

n this chapter some notions of classical integrability following the Liou-ville’s theorem are reviewed. We analyze the connection between the inte-grability of classical systems and the ergodic theorem which guarantees the validity of the ensembles technique. Furthermore the problems arising when a quantum system is considered are discussed. In particular, we follow some recent works ([37] or [42, chap. IV]) where it is shown a possible definition of quantum integrability. Although it is not intuitive, such a definition will be useful to give a first classification between integrable and non-integrable systems.

2.1

Classical Integrability

In order to show the precise mean of integrability in classical mechanics we will use a hamiltonian approach. Let us remind some basic facts that will be the background of our arguments1. Given a generic system described by the

hamil-tonian H(pj, qj), where pj and qj is a set of conjugate variables, it is possible to

characterize its dynamics by solving a system of coupled differential equations called the Hamilton’s equations:

   ˙qj = ∂H∂pj ˙pj = −∂H∂qj where ˙q = dq dt and ˙p = dq dt. (2.1) For a system with n degrees of freedom we can introduce a 2n × 2n matrix J:

J = 0 I

−I 0 !

, (2.2)

such that the hamilton’s equations can be written as: −

˙x = J · ∇H,

1The notions exposed here can be found in all the classical books of mechanics. See for

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CHAPTER 2. INTEGRABILITY 19 where ∇H = (∂H ∂pj, − ∂H ∂qj) and − →x = (p, q).

The matrix J can be used to define the Poisson brackets on the space composed by differentiable functions defined on the 2n−dimensional phase-space M:

{F, G}= (∇F, J · ∇G) = N X i=1 ∂F ∂pi ∂G ∂qi∂F ∂qi ∂G ∂pi  . (2.3) A hamiltonian system with n degrees of freedom is therefore characterized by a 2n-dimensional phase space, a Poisson structure and a Hamiltonian. For any function G(p, q) defined on the phase space, it can be shown that the time evo-lution is given by the Poisson brackets with the Hamiltonian:

dG(p, q)

dt = {H, G}. (2.4)

Since, trivially, {H, H} = 0, the hamiltonian is a conserved quantity and then the trajectories on the phase-space of the system live on a manifold with constant en-ergy. The hamiltonian H(−x) can be expressed in terms of other variables H(−x0) provided that the −→x and the −→x0 are related trough a canonical transformation, that is

− →x0

= A−x with AJAt= J (2.5) where A is 2n × 2n matrix implementing the transformation and −x and −→x0 are 2n-dimensional vectors of the type defined previously. Since the matrix J is non-degenerate the inverse, J−1 = −J exists. Using this property it is possible to

define a measure on the phase space by introducing the bilinear ω(−x , −y): ω(−→x , −y) = (−→x , J−1−→y), (2.6) where (, ) denotes the scalar product. It is easy to show that the measure defined in this way is invariant under canonical transformations, indeed:

ω(A−x , A−y) = (A−x , J−1A−y) = −(−→x , AtJ A−y) = (−→x , J−1−→y) = ω(−x , −y). (2.7) The Lioville’s theorem states that a classical system with n degrees of freedom described by a Hamiltonian H(−x) is integrable if we can find n functions Fi

defined smoothly over the phase-space M such that:

• the Fi are conserved charges, thus, for every i = 1, . . . , N one has

{H, Fi}= 0;

• the Fi are functionally independent of each other;

• all the Fi are in involution:

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CHAPTER 2. INTEGRABILITY 20 This definition of integrability is sufficient for our purposes, anyway it can be improved. A more formal statement of the Liouville’s theorem can be found for example in [34]. If a system is integrable, a particular type of conjugate vari-ables can be introduced. These varivari-ables are called action-angle varivari-ables and the differential equations governing the dynamics can be integrated by quadra-tures [35]. The solution in terms of the action-angle variables displays a periodic motion on invariant tori in the phase-space. This periodicity is typical of inte-grable systems and it is one of the most important differences between classical integrable and non-integrable models. Indeed, if the evolution of the system in the phase-space is not chaotic, the ergodic theorem does not hold and the average values of the various quantities will depend on the particular orbit of the system in phase-space. Therefore, since the points of the sub-variety of the phase-space with constant energy are not equivalent, the ensemble technique fails.

2.1.1 Integrability and ergodicity

Let us consider a system of N particles described by a certain hamiltonian H(p, q) that defines a sub-variety of constant energy in the phase-space on which the orbit of the temporal evolution lives. The ergodic hypothesis states that, for every observable F (p, q) defined on the subvariety H(p, q) = E, the late-time average can be calculated averaging over an infinite number of copies of the system satisfying the only requirement of fixed energy. This picture coincides with the Gibbsian ensemble technique [2] discussed formerly and that it is formalized by the introduction of a density distribution defined in the phase-space ρ(p, q, t) such that

ρ(p, q)d3qd3p= number of copies contained in the volume d3qd3p. (2.8) The ensemble average of the quantity F (p, q) is defined as

hF(p, q)i = R

d3Npd3NqF(p, q)ρ(p, q)

R

d3Npd3N(p, q) . (2.9)

The ergodic condition can be thus rewritten as lim T →∞ 1 T Z T 0 dtF(q(t), p(t)) = R d3Npd3NqF(p, q)ρ(p, q) R d3Npd3N(p, q) , (2.10)

where q(t), p(t) ∈ Γ(p, q) and Γ(p, q, ) is the trajectory of the system in the phase-space. One fundamental assumption behind the ergodic condition and the validity of the Gibbs ensemble technique, as sketched in the last paragraph, is that all the copies of the system must be equiprobable. This statement can be rephrased as: the points satisfying the condition of constant energy must be “touched” at least once by the path representing the time evolution of the system in the phase-space. In fact the averaging procedure over the ensemble takes in account all the points compatible with the requirement of fixed energy, therefore if some of them do not concur at the time evolution Eq. (2.10) does not hold. Those systems whose trajectories display a certain regularity can not be treated as ergodic ones. In this category both integrable and quasi-integrable systems fall.

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CHAPTER 2. INTEGRABILITY 21 Quasi integrable systems are distinguished by the property that are very “close” to be integrable, examples of this kind are integrable systems subjected to weak non linear perturbations. These systems, although strictly speaking are non-integrable, display an absence of ergodicity because the trajectories in the phase-space are quasi periodic and, in the infinite time limit, do not cover all the allowed sub-variety in the phase-space2. Non-integrable systems are instead qualitatively

different. Indeed in this case the evolution is “chaotic” and as time passes the phase-space on which the system lives is completely explored and thus the relation (2.10) is valid.This means that for “random” time evolution all the copies of the system that preserves the condition of energy conservation are de-facto equivalent and then we have ergodicity.

What has been reported here about the ergodic problem is only partial, indeed we avoid to mention a lot of details and problematics that are behind the concept of ergodicity, but what has been discussed should be sufficent to give an overview and to remind some important concepts on which all the classical statistical physics (and not only) is based.

2.2

Quantum Integrability

The aim of this paragraph is to introduce the reader to some concepts relating the question of quantum integrability and quantum integrable systems. Our discussion will follow what reported in the works [42, chap. IV] or [37] and [38]. First of all, we must point out that in this thesis, as reported in chapt. 4, we consider a post-quench hamiltonian which can be mapped to a free fermionic one, and the initial state was prepared in the ground state a free bosonic hamiltonian. Therefore we are dealing with integrable models since, no matters what kind of definition of quantum integrability we adopt, free theories should fall into the integrable class. The first naive tentative that one can do in order to give a proper definition of quantum integrability is translating in “quantum language” the Liouville’s theorem that holds for classical systems. Then, if we replace the Poisson brackets with the commutators under the prescription

{, } → i

~[, ], (2.11)

we can say that:

Definition 1. A system is quantum integrable if we can find a complete set of independent commuting operators (i.e. [Im, In] = 0 ∀m, n = 1, . . . , dim(H)).

This definition, although seems to be quite reasonable, shows some pitfalls. The first question is that a classical definition transported in quantum mechanics could be ill-conditioned by the fact that the degrees of freedom are counted in a total different way in the two cases. In classical mechanics the number of degrees of freedom is just the number of couples of conjugate variables that one needs

2The first and perhaps most famous demonstration of quasi periodic motion for quasi

inte-grable systems is the so called FPU (Fermi, Pasta, Ulam) problem, all the details can be found in reference [36]

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CHAPTER 2. INTEGRABILITY 22 to specify the dynamics, instead in quantum mechanics the number of degrees of freedom of a system is just the dimensionality of its Hilbert space. For this reason the requirement of completeness it has been added in the definition, in other words: the cardinality of the conserved charges has to be the same as the dimensionality of the Hilbert space H. The second, and more severe, problem is that according to the definition 1 every quantum system with finite-dimensional Hilbert space would be integrable. In fact by the spectral theorem every her-mitian Hamiltonian is diagonalizable, then one can define dim(H) commuting operators, Im = |ψmihψm|. This is not acceptable if we want a classical

corre-spondence, therefore we must abandon the idea of translating “word by word” the integrability condition from classical to quantum language.

Another commonly definition used in literature is:

Definition 2. A quantum system is integrable if its full set of eigenstates can be constructed, i.e. if it is exactly solvable.

Also this definition could be satisfactory at first sight, especially because reminds the classical feature of angle-action variables. But this sounds as a consequence of integrability rather than as a definition. Furthermore it sounds too simplistic because it is not specified if models solvable with different methods should display different physical behavior.

Before proceeding, we follow what done in [37] and itemize some inescapable requirements that we want to be fulfilled by a proper definition:

• it should be unambiguous;

• it should define different classes to which different models belong; • all the classes should display different physical behavior.

We realize that without a general and rigorous definition it is very hard to fulfill all these criteria. In particular let us note that the second definition we gave (that at the moment is the most reasonable one) fails in the third criterion. Another subtle point concerns the number (i.e. the cardinality) of the conserved charges. In the first definition, we required the completeness in the sense that the cardi-nality of the conserved charges should be the same of the dimensiocardi-nality of the Hilbert space. A theorem by Von Neumann [38] states that: given a set of com-muting operators {Im}, it is possible to construct another operator I such that

every Im can be viewed as a functions of I, Im = fm(I), so the number of the

conserved charges can be ill-defined. Furthermore the request of completeness is often accompanied by the request of locality (see last chapter 1.3.1), but it is not clear what should mean: “a complete set of local charges”, since the “locality” requirement restricts the number of possible operators. The confusion probably arises from the misunderstanding between the concept of infinite and the concept of complete set of charges [37].

A possible escape from all these problematics has been proposed by J.S. Caux and J.J. Mossel and is reported in [37] and [42]. The definition they gave is a very formal one but fulfills all the criteria and, until now, seems to work, contrarily to the previous ones we considered. It is worth to introduce some preliminary

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CHAPTER 2. INTEGRABILITY 23 concepts. Let us define a size sequence as a strictly increasing string of inte-ger numbers, (N1, . . . , Nm, . . .) with N1 < N2 < N3. . . etc. To each number

in the string it is associated a Hilbert space H(Ni) obtained by tensoring a

fi-nite dimension “basic” Hilbert space (with dimHi = di ) Hi Ni times, so that

dimH(Ni)

 =QNi

j=1dj ≡ dNi is also finite. Higher dimensional Hilbert spaces in

the sequence are built following the simple rule: HNi+1 = HNi⊗ H

j. In the basic

Hilbert spaces Hj self adjoint operators can be represented by dj× dj hermitian

matrices that can be decomposed in a chosen basis, therefore all the operators acting on the tensored higher dimensional Hilbert space HNi can be decomposed

as: O(Ni)= X i1,...,iNi O(Ni) i1,...,iNie i1...iNi, (2.12) where ei1···Ni is the basis obtained tensoring N

i times the chosen basis of the

fundamental Hilbert space Hi. The number of non-zero entries in (2.12) is

de-noted as Ne



O(Ni). Considering a size sequence of operators (O(N1), O(N2). . .) it is defined the density character of O(Ni) as the “nature” (i.e. linear,

polyno-mial, exponential, etc. . . ) of the minimal function f(Ni) such that Ne



O(Ni)<

f(Ni). At this point it is possible to introduce a size sequence of Hermitian

operators (H(N1), H(N2), H(N3). . .) that can be interpreted as Hamiltonias act-ing on the respective Hilbert spaces. Since these operators are hermitian and finite-dimensional, automatically possess complete sets of conserved charges Q(Ni)

m

which can be arranged in such a way that several size sequences can be defined: (Q(Ni)

m , Q(Nmi+1), Qm(Ni+2). . .) for all m = 1, . . . , mmax ≤ d(Ni).

After these preliminary concepts we are ready to report the definition.

Definition 3 (Caux & Mossel). A Hamiltonian of density character O (f(N)) is quantum integrable if it is a member of a sequence (H(N1), H(N2), H(N3). . .) of operators, having O (f(N)) density character in a certain basis too, for which it is possible to define a sequence of sets of operators ({Q(N1)}, {Q(N2)}, {Q(N3)}, . . .) such that:

• all operators QNi

m in {Q(Ni)}commute with each other and with the relative

hamiltonian H(Ni);

• the operators in {Q(Ni)} are algebraically independent;

• the cardinality of the set {Q(Ni)} becomes unbounded in the infinite size

limit;

• each member of a set {Q(Ni)}can be embedded within a sequence of

oper-ators (Q(Ni)

m , Q(Nmi+1), Q(Nmi+2). . .) with O (f(N)) in a given basis.

This definition is very formal and at first sight hard, anyway it fulfills all the required criteria; in particular it can be done a partition between different classes of integrable models according to their density character O (f(N)).For all the details see [37]. Let us note in particular that, by definition 3, free discrete

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CHAPTER 2. INTEGRABILITY 24 models are linear quantum integrable in the sites basis (i.e. of density character O(N) where N is the number of sites)3.

3

Let us note that in the continuum case this means that we have an infinite number of conserved charges, indeed any continuum theory can be obtained from a lattice regularization in which subset of conserved charges can be chosen O(N ).

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CHAPTER

3

THE LIEB-LINIGER MODEL

I

n this chapter the model of our interest is introduced. We will show that it is solvable using the coordinate Bethe ansatz and we will outline how to find the general solution in the periodic case. Furthermore, the limit of “hard-core” bosons is discussed in detail and we show, in two different ways, that in this limit the model is solvable, since it can be mapped to a free fermion theory. Finally, the last part of the chapter is devoted to the discussion of some important results for a quench from non-interacting to strongly interacting bosons with PBC, without confining potential following [47].

3.1

The model

The Lieb-Liniger model is an example of interacting model and describes one di-mensional bosons with a contact interaction. It was introduced and solved in 1963 [39] and it gives a good description for bosons in effective 1D lattices, especially when the limit of strong interaction is considered [40, sec. 5]. Exact solutions can be found both for attractive and repulsive interaction. The hamiltonian in first quantized notation for a system composed by N particles is:

HN = N X i=1 2 ∂x2i + 2c X j>i δ(xi− xj), (3.1)

where we set ~ = 2m = 1, the constant c represents the interaction strength and we will always consider the repulsive case (i.e. c > 0). The aim in this paragraph is to find the eigenfunctions χN(x1. . . xN) of (3.1):

HNχN(x1. . . xN) = EN(k1. . . kN)χN(x1. . . xN). (3.2)

Since we are dealing with a bosonic system we must impose the symmetry of the functions χN(x1. . . xN) under the exchange of the coordinates, that is:

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CHAPTER 3. THE LIEB-LINIGER MODEL 26 for every couple (i, j).

So far we have not discussed about boundary conditions, these are relevant since they can drastically change the wave function especially when working with finite N. Different boundary conditions can be implemented: we can study a one dimensional bosonic gas lying on a ring of circumference L, (this coincides with the imposition of periodic boundary conditions) or we can consider a box of length L, meaning that we are fixing the boundaries. The model can also be solved in the most general case of twisted boundary conditions. The effects of the boundaries are expected to vanish when the limit L → ∞ is considered. Since our purpose is to give only an outline of the methods used in finding the exact solutions of (3.1), we will consider the easiest case of periodic boundary conditions.

The wave function has to be symmetric in the spatial coordinates, therefore we can consider only the following domain

D: x1 < x2< · · · < xN, (3.3)

in D the (3.1) reduces to a free Hamiltonian H0N = N X i=1 2 ∂x2i, (3.4)

with the following constraint:

 ∂xi+1 ∂xi − c  χN = 0, xi+1= xi, (3.5)

imposed by the δ interaction. In order to understand why the constraint (3.5) works, let us show how it arises when we have only two particles.Tthe general-ization for N bosons is straightforward. Introducing the variables

(

z= x2− x1

Z = x2+x1

2

the Hamiltonian (3.1) can be written as: H2 = −2

2

∂z2 −

2

2∂Z2 + 2cδ(z), (3.6)

if we now integrate equation (3.2) for the Hamiltonian (3.6) in the variable z over an interval of measure   1 (i.e. −

2 < z <  2), we obtain:  −2 ∂zχ2 2 2 + 2 ∂Z2χ2 ! + 2cχ2= O(). (3.7)

In the limit  → 0 (3.7) becomes exactly (3.5):  ∂x2 − ∂x1 − c  χ2 = 0, x2= x1, (3.8)

this means that, if the wave function χ2(x1, x2) is an eigenfunction of the

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CHAPTER 3. THE LIEB-LINIGER MODEL 27 Before considering the general solution of (3.1) with the boundary conditions (3.5), let us see what happens in the simplest possible case: N = 2. The wave function in this case can be written as:

χ2(x1, x2) = C1eik1x1+ik2x2 + C2eik2x1+ik1x2, (3.9)

where the coefficients C1 and C2 are determined by the constraint (3.8):

(ik1− ik2− c)C1 = (ik2− ik1− c)C2, (3.10) C2 C1 = −e iφ(k1−k2) φ(k 1− k2) = −i ln c+ i(k1− k2) c − i(k1− k2) , (3.11) the function φ(k1− k2) can be thought as a phase-shift due to the contact

inter-action. The general symmetric solution is:

χ2(x1, x2, k1, k2) = sgn(x2− x1)[eik1x1+ik2x2−isgn(x2−x1)φ(k1,k2)/2

−eik2x1+ik1x2+isgn(x2−x1)φ(k1,k2)/2]. (3.12)

For a generic number of particles N we can proceed in the following way 1. We

make the ansatz that the solution can be written as a superposition of plane waves

ψ(x1. . . xN) =

X

P

APeikP1x1+···+ikPNxN, (3.13)

where ki are the (quasi)momenta, P is a permutation of the ki and the sum is

extended over all the permutations. All the possible permutations are done by exchanging two possible indices at a time, therefore the coefficients AP can be

found exploiting the results from the two particles case AP

AP0 =

i(ki− kj) + c

i(ki− kj) − c = −e

iφ(ki−kj), (3.14)

where ki and kj are the momenta exchanged by the permutation. Therefore we

obtain [39]:

AP = C(−1)P

Y

j<i

(kPj− kPi+ ic) (3.15)

where C is a normalization constant. The wave functions that we found in this way are eigenfunctions of the Hamiltonian (3.4) with eigenvalue

E =

N

X

j=1

kj2. (3.16)

They are also eigenfunctions of the momentum operator, with total momentum P = N X i=1 ki. (3.17) 1

A part from the original reference [39], there are several reviews about the coordinate Bethe

Ansatz techniques used in the solution of the Lieb-Liniger model, e.g. [41] or [42, chapt. 2].

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CHAPTER 3. THE LIEB-LINIGER MODEL 28 So far we said nothing about the permitted values of the quasi-momenta ki; these

values must be found imposing the boundary conditions. As discussed formerly we will consider periodic boundary conditions (PBC). If the particles lying in a length L, PBC are imposed if for every 1 ≤ j ≤ N we have:

ψ(x1. . . xj. . . xN) = ψ(x1. . . xj + L . . . xN), (3.18)

this effectively means that our system has the geometry of a ring. If a particle moves around the circle it will acquire a phase due to two different contributions. The first contribution will be the dynamical phase (i.e. eikjL). The second

contribution will come from the scattering with all the others particles around the ring, when PBC are considered. The sum of these contributions is a integer multiple of 2π [41]. More formally we can say that if we take two permutation of the momenta2 such that the difference between the two(P and P0) is the shift of

one index (i.e. P0

N = PN −1and P

0

1= PN. . . ), the following condition must hold

AP0

AP = e

ikPNL, (3.19)

this is equivalent to:

eikjL=Y j6=i kj− ki+ ic kj− ki− ic ! , j= 1, . . . , N. (3.20) The (3.20) is a set of N equations for N variables (the quasi-momenta ki) called

Bethe equations. Although the number of equations is the same of the number of variables the solution is not unique since different solutions can be shifted by integer multiples of 2π. This fact is evident is we take the logarithm of (3.20):

kjL= −i X i6=j +X i6=j −iln c+ i(ki− kj) c − i(ki− kj) ! + 2πIj, (3.21) that is : kjL= (N − 1)π + X i6=j φ(ki− kj) + 2πIj. (3.22)

The numbers Ij are integers if N is odd, half-integers otherwise. From equation

(3.22) it is clear that the sum of the dynamical phase and the phase acquired due to the interaction has to be a integer multiple of 2π. Let us note that the set of integers Ij are quantum numbers that identify the state of the system. It is not

hard to convince ourselves that the ground state is the one in which the integers Ij are symmetrically distributed around the zero

Ij = −

N + 1

2 + j j = 1 . . . N. (3.23) This fact can be proved in different ways but we will omit the demonstration. It is important to remark that the Bethe solution of the Lieb Liniger model provides all the quasi-momenta to be distinct, indeed if we have ki = kj for a given (i, j)

the wave function vanishes. In particular, this last point shows that, also if we are considering bosons in real space, the contact interaction reveals a “fermionic nature” in momentum space.

2

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CHAPTER 3. THE LIEB-LINIGER MODEL 29

3.2

Tonks-Girardeau limit

The limit for c → ∞ in the Hamiltonian (3.1) is usually referred as “Tonks-Girardeau” limit since it was solved for the first time by Girardeau [43] and, in its classical version by Tonks [44]. In this regime the Hamiltonian (3.1) describes a one-dimensional system of strongly interacting bosons. What we calculated so far for the generic Lieb-Liniger gas is still valid and the solution for the Girardeau gas could be found considering, with a certain care, the limit for c → ∞ of the previous results. However we will derive the correct solution focusing on the physical meaning of the limit considered.

Imposing a strong repulsive contact interaction means that the probability for two particles to be “close” to each other is very low. For c → ∞, when deal-ing with point like particles3, this is equivalent to say that, given a many-body

eigenfunction of the hamiltonian χN(x1. . . xN), the following condition must be

respected:

χN(x1. . . xi. . . xj. . . xN) = 0 if xi = xj, (3.24)

for every couple (i, j). Equation (3.24) sounds like a fermionic requirement. Effectively in this regime, since it is not allowed multiple spatial occupation, the bosonic gas is physically undistinguishable from a free fermionic one, as will be clearer soon. The Hamiltonian for a Girardeau gas can thus be written as:

HT G= N X i=1 2 ∂x2i, (3.25a) χN(x1. . . xi. . . xj. . . xN) = 0 if xi = xj, (3.25b)

let us note that relations (3.25) are, as expected, exactly the same as (3.4) with the constraint (3.5) in the limit c → ∞. A possible solution to (3.25) is a determinant of a N × N matrix with entries eikixl (i.e. a Slater determinant).

However we must take into account the fact the we are still dealing with a bosonic system, therefore the wave function should be symmetric under the exchange of two spatial coordinates. Following [43] the wave function is:

χN(x1. . . xN|k1. . . kN) = CN!det[exp{iklxj}] Y i<j sgn(xl− xj), (3.26) that is: χN,G(x1, x2, . . . , xN) = Y i<j sgn(xi− xj)χN,F(x1, x2. . . , xN), (3.27)

where χN,F(x1, x2. . . , xN) is a many-body eigenstate of a free fermionic

hamil-tonian. The allowed values for the momenta are found, as usual, imposing the boundary condition. Following what has been done for the general Lieb-Liniger model, (i.e. imposing PBC) we find

eikjL= (−1)N −1 for j= 1, . . . , N. (3.28)

3

A more realistic physical situation would prescribe to consider the particles as spheres of finite ray, this has been done classically and is fully described in [44].

Riferimenti

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